SUPPLEMENTARY INFORMATION
Determination of the overall photon fluence and the apparent photon fluence-based rate constant
for a photochemical process in solution
According to the Second Law of Photochemistry, the photochemical processes at various wavelengths
are independent from each other and additive. Therefore, the overall kinetics describing the decay of
compound B under polychromatic radiation is expressed by Eq. 6c:
æ [B] ö
ln ç 0 ÷ = k1t
è [B]t ø
(S-1a)
where k1 is obtained from Eq. 6d. Note that the ln removal is the integral of time-based rate constants
over the wavelength range, multiplied by the exposure time. In Eq. S-1a only the rate constant is
wavelength-dependent (i.e., the variable t, is not).
Over the same time t in Eq. S-1a, a certain photon fluence Fp,o (water) is delivered to the water. The
effect is the same, that is, the same ln removal is observed. Therefore
æ [B] ö l2
æ [B]0 ö
0
ln ç
= k1t = ln ç
÷ = ò k1,Fp,o Fp,o,l (water) d l
÷
ç
÷ l
[B]
[B]
è
t ø
F
è
1
p.o ø
(S-1b)
where k1,Fp,o is the photon fluence-based rate constant (m2 einstein–1) at wavelength  and Fp,o,l (water) is
the spectral photon fluence (einstein m–2 nm–1) contribution at wavelength .
Unlike the time-based rate constants at various wavelengths, the photon fluence-based rate constants
at various wavelengths are not additive to an overall polychromatic photon fluence-based rate constant,
since they are linked to a parameter (photon fluence at a respective wavelength) that depends on
wavelength. Equation S-1b obeys the Second Law of Photochemistry, that is, the ‘additive’ property of
photochemical events.
Since an overall photon fluence can be calculated based on spectroradiometer data, and the ln
removal is the same for a given exposure time (i.e., photon fluence), one can determine an apparent
polychromatic photon fluence-based rate constant k1,F
app
p,o
l2
òl k
1,Fp,o
(l )Fp,o,l (water) d l = k1,F
p,o
app
Fp,o (water) = é k1,F app Ep,o (water) ù t
ë p,o
û
(S-1c)
1
where Fp,o (water) is the overall photon fluence delivered to the water across the wavelength range
considered from the light source emission spectrum and the absorption spectrum of the photoactive
substance.
From Eqs. S-1b and S-1c
k1,F
app
p,o
=
k1
Ep,o (water)
(S-2a)
On inserting Eqs. 12a and 15b into Eq. S-2a
l2
k1,F
app
p,o
=
òF
ln(10) l
B
( l )e B ( l )Ep,0 l ´ RF(l ) ´ WF(l )d l
(S-2b)
1
10
l2
òl éë E
0
p,l
´ RF(l ) ´ WF(l ) ùû d l
1
Again if decremental wavelength bands are used (e.g., 1 nm width), Eq. S-2b may be integrated
using the trapezoidal rule.
In Eqs. S-2, if the spectral irradiances are given in mW cm–2 nm–1, the factor of 10 in the rate
constant expressions should be omitted, since 1 mW cm–2 = 10 W m–2.
If the ‘apparent’ quantum yield F B (l )app is independent of wavelength, since, in principle, all the
other parameters are known, F Bapp can be determined from a rearrangement of Eq. S-2b
l2
F Bapp =
10k1,F
ò éë N l ´ RF(l ) ´ WF(l ) ùû d l
app
l1
p,o
ln(10)
l2
òe
l1
B
(S-3)
( l )N l ´ RF(l ) ´ WF(l )d l
Eq. S-2b differs considerably from the approach taken by Stefan and Bolton (9), since photon-based
units have been used instead of energy-based units.
Excel spreadsheet to give an example of how to calculate the average irradiance
An Excel spreadsheet (ExampleCalculationOfAverageIrradiance.xlsx) is available in the Supplementary
Information. This spreadsheet provides an example of how to calculate the average irradiance using
Eq. 12b.